This is a magic trick performed by two magicians, A and B, with one regular, shuffled deck of 52 cards. A asks a member of the audience to randomly select 5 cards out of a deck. the audience member -- who we will refer to as C from here on -- then hands the 5 cards back to magician A. After looking at the 5 cards, A picks one of the 5 cards and gives it back to C. A then arranges the other four cards in some way, and gives those 4 cards face down, in a neat pile, to B. B looks at these 4 cards and then determines what card is in C's hand (the missing 5th card). How is this trick done?

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Note 1: There's no secretive message communication in the solution, like encoded speech or ninja hand signals or ESP or whatever ... the only communication between the two magicians is in the logic of the 4 cards transferred from A to B. Think of these magicians as mathematicians.

Note 2: This magic trick is originally credited to magician and mathematician Fitch Cheney.

When B gets the packet of four cards, he must determine which of the remaining 48 cards was chosen by C but left out by A. The cards he sees can only be put into 4! = 24 different ways (according to their values high to low prearranged to represent the 1st possible card, second, etc.) But this is only half of what is needed. We need only one more bit (literally, binary digit) of information.

Unfortunately, A must place the cards face down. That prevents him from encoding that extra bit by presenting them face up or face down to represent either the first 24 possible cards or the last 24 possible cards.

If the cards have asymmetric designs on the back, then information (one bit's worth) could be encoded by either aligning them all or not aligning them. But presumably the use of asymmetric backs (say pictures of dogs, rather than checkerboard designs) would be akin to ninja hand signals.

Presumably the true method consists of devising a scheme of specifying beforehand what card would be chosen to be left out, out of any given hand, so that only 24 or fewer possibilities remain, rather than 48.

Posted by Charlie on 2009-05-29 12:07:50